Clean living, it would seem, really does keep a body healthy. Just look at those nonsmoking, nondrinking boys in “The Book of Mormon.” They’ve been hanging around in the insalubrious, soul-sapping environs of Broadway for more than three years. Yet they don’t look a day older than when they arrived.

On my fourth visit to “The Book of Mormon” on a recent night, the show still had me at “Hello,” the first word of the first song in this long-running musical. I had anticipated a certain falling off of religious fervor and discipline in this ebullient satire about naïve Mormon missionaries in Uganda, written and composed by Trey Parker, Robert Lopez and Matt Stone, directed by Mr. Parker and Casey Nicholaw, also its choreographer. Three years is a long time for a peppy musical to stay peppy.

But gosh darn it if I didn’t feel born again all over again, ready to spread the word about the faith that this show preaches and, more important, practices. I do not mean the religion of the followers of Joseph Smith, which is so scrupulously disassembled here, but the holy faith of musical comedy, into which my mama baptized me before I could walk.

Much of this show’s enduring freshness comes from its unabashed identity as an archetypal musical. It’s a celebration of the spirit that moves otherwise sane human beings to accept the absurdity of people setting their cares and woes to sprightly song. As for the chorus of the Mormon lads who begin the production ringing doorbells to spread the word, the most important quality for this show’s cast is an unqualified belief in the power of what it’s selling.

Well, that and the ability to perform precision-tool dancing, to sing on key and to perceive the stage as a snark-free zone, even when extolling Disney World as an earthly paradise or the bliss of having your own personal planet in the afterlife. If the audience wants to hoot at such declarations, it is welcome to.

But those onstage must project an unflinching sincerity and optimism. These are traits that come more easily to the young, and most of the performers here are indeed on the sunny side of 40.

“Mormon,” at the Eugene O’Neill Theater, allows for easier wiggle room in its principal roles than musicals that rely on signature star power. (“The Producers” never fully recovered from the departure of Nathan Lane and Matthew Broderick.) As Ugandan villagers who hear the call of the gospel, the Broadway veterans Nikki Renée Daniels and Daniel Breaker are so light-handedly charming they never feel like second choices. Nor does Matt Loehr, who brings a comfy touch of Paul Lynde to the part of the sexually self-repressing Elder McKinley.

The starring roles of the Mutt and Jeff missionary team originally played by Andrew Rannells and Josh Gad have been seamlessly retailored by Nic Rouleau and Ben Platt. They’re a little looser than their predecessors, which means that Mr. Rouleau’s Elder Price isn’t as blindingly confident as Mr. Rannells’s was; he’s less galvanizing, but easier to root for.

Mr. Platt’s sensational Elder Cunningham is a hapless ironist, who keeps assuming and discarding the voices he’s heard on television because he doesn’t trust his own. When he finally sees the light, and acquires confidence, it’s the light of his own made-up make-believe.

The transformation feels just right, because “The Book of Mormon” is about the triumph of faith in fantasy. As performed by Elder Price, “I Believe” may be a hysterically comic number. But it is also an anthem to the spirit that keeps that this show alive and well.

Correction:

A picture caption that ran with an earlier version of this article misidentified the character portrayed by Nic Rouleau in “Book of Mormon.” He plays Elder Price, not Elder Cunningham.